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On the new k-th Yau algebras of isolated hypersurface singularities

  • Naveed Hussain
  • Stephen S.-T. YauEmail author
  • Huaiqing Zuo
Article
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Abstract

Let V be a hypersurface with an isolated singularity at the origin defined by the holomorphic function \(f: ({\mathbb {C}}^n, 0)\rightarrow ({\mathbb {C}}, 0)\). The Yau algebra L(V) is defined to be the Lie algebra of derivations of the moduli algebra \(A(V):= {\mathcal {O}}_n/(f, \frac{\partial f}{\partial x_1},\cdots , \frac{\partial f}{\partial x_n})\), i.e., \(L(V)=\text {Der}(A(V), A(V))\) and plays an important role in singularity theory. It is known that L(V) is a finite dimensional Lie algebra and its dimension \(\lambda (V)\) is called Yau number. In this article, we generalize the Yau algebra and introduce a new series of k-th Yau algebras \(L^k(V)\) which are defined to be the Lie algebras of derivations of the moduli algebras \(A^k(V) = {\mathcal {O}}_n/(f, m^k J(f)), k\ge 0\), i.e., \(L^k(V)=\text {Der}(A^k(V), A^k(V))\) and where m is the maximal ideal of \({\mathcal {O}}_n\). In particular, it is Yau algebra when \(k=0\). The dimension of \(L^k(V)\) is denoted by \(\lambda ^k(V)\). These numbers i.e., k-th Yau numbers \(\lambda ^k(V)\), are new numerical analytic invariants of an isolated singularity. In this paper we studied these new series of Lie algebras \(L^k(V)\) and also compute the Lie algebras \(L^1(V)\) for fewnomial isolated singularities. We also formulate a sharp upper estimate conjecture for the \(\lambda ^k(V)\) of weighted homogeneous isolated hypersurface singularities and we prove this conjecture in case of \(k=1\) for large class of singularities.

Keywords

Isolated hypersurface singularity Lie algebra Moduli algebra 

Mathematics Subject Classification

14B05 32S05 

Notes

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  • Naveed Hussain
    • 1
  • Stephen S.-T. Yau
    • 1
    Email author
  • Huaiqing Zuo
    • 1
  1. 1.Department of Mathematical SciencesTsinghua UniversityBeijingPeople’s Republic of China

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